For requesting, clarifying, and comparing definitions of mathematical terms.

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17 views

Definition of normal extension.

Let $E/F$ be an algebraic extension. Then, $E/F$ is normal iff $E$ is a splitting field of a family of polynomials in $F[X]$. So does this mean that if $E$ is a splliting field of a given ...
4
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2answers
143 views

Evaluating a definite integral by definition

I have an area function $A(x)$ defined as $$A(x) = \int_{-1}^{x} (t^2 + 1)\space dt$$ ... and I would like to use the definition of definite integral to evaluate it. I started this way $$A(x) = ...
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1answer
65 views

Why does this author define cardinality indirectly?

I'm studying Enderton's Elements of Set Theory and in the page 129 he defines what it means two sets being equinumerous: After that in the page 136 he defines cardinality: Why doesn't he define ...
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2answers
57 views

Two definitions of Jacobson Radical

I have in my notes that the Jacobson radical of a ring $R$ is: $J(R) = \cap${$I$ | $I$ primitive ideal of $R$} $= \cap$ {$Ann_R M$ | $M$ simple $R$-module}. I have now seen elsewhere that $J(R) = ...
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1answer
51 views

What does the notation $\mathbb R[x]$ mean?

What does the notation $\mathbb R[x]$ mean? I thought it was just the set $\mathbb R^n$ but then I read somewhere that my lecturer wrote $\mathbb R[x] = ${$\alpha_0 + \alpha_1x + \alpha_2x^2 + ... + ...
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1answer
36 views

Is this alternate definition of Limit correct?

Would it be correct to define the limit of a series as the smallest number that no number in the series is greater than (for an increasing series, the other way around for a decreasing series, i.e the ...
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1answer
39 views

How do I prove cardinality is well-defined?

I define equinumerous and cardinality in this way: $A$ and $B$ are equinumerous (written $A\sim B$) if there is a bijection between them. We say $card(X)=card(Y)$ if $X\sim Y$. I would ...
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13 views

Defining the rational function field in n variables.

Reading over an editing my dissertation "Elementary functions" and i am having trouble with my definition of a rational functions in n variables, this is what i have written but its missing one part: ...
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1answer
49 views

Vector Space: is there a general definition of annihilator beyond the dual space?

The definitions I see for the annihilator of a subset S of a vector space V over a field F is the subset $S^0$ of the dual space $V^*$given by $S^0 = \{\varphi\in V^\star:\varphi(S)=0\}$, and the ...
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1answer
41 views

Does the line y=mx have equal intercepts?

This line passes through the origin and has zero intercepts. Can this be called as a line having equal intercepts? What is the definition of intercepts of a line then?
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1answer
113 views

Why does 2+2 equal to 4? [duplicate]

The question is in the title. I am very appreciative of any time and concern put into belaboring this relatively little problem.
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1answer
30 views

Riemann integrable function over bounded set

In my calculus course we extended the definition of riemann integrable to functions whose domain are jordan-measurable sets, but can we extend the definition if we just ask for the domain to be ...
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2answers
77 views

Definition of Aut(G) in the graph theory and group theory

For a fixed group G, we define the collection of group automorphisms is the automorphism group Aut(G) in the group theory. (An automorphism: a permutation on the set G) In the graph theory, on the ...
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92 views

Funny translations of mathematical words [closed]

As already noticed in this question there are some mathematical words that literally translated from a language to english (or from english to this language) means something totally different. A few ...
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1answer
83 views

How to translate math technical terms?

What is a good way to translate mathematical technical terms? This can sometimes be hard because some words have different meanings in some language. For example: "eigenwert" (= "eigenvalue" ...
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1answer
149 views

Is my attempt to define the concept “smooth manifold” as a structure satisfying certain axioms correct?

In the lecture notes for a class I'm currently taking, smooth manifold structures are defined as equivalence classes of atlases. However, the issue I'm having is that its not entirely clear (to me) ...
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1answer
18 views

Radical function in two (or more) variables

I'm reading a paper where the author uses the word radical function for a function $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$. I understand the definition of a radical function if $n=1$, but what if ...
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1answer
75 views

Complex number field: ''essentially'' unique?

I solved the following exercise but have trouble making sense of the result: If $\widetilde{\mathbb C}$ is another field of complex numbers and $\varphi : \mathbb C \to \widetilde{\mathbb C}$ is a ...
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1answer
40 views

Useful definition of limit?

Is the following definition of a limit a useful one? Does it make sense? Why/Why not? $lim_{x\rightarrow a} f(x) = b \Leftrightarrow \forall \varepsilon > 0 : 0<|x-a|<\varepsilon ...
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27 views

Finding the Partial Derivative of this function (Chain Rule)

This problem is not part of the assigned problem set, but I would really like to figure it out. I spend the last 2 days thinking about it and stuck on the chain rule part. I know how to do this using ...
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2answers
45 views

Is the empty family of sets pairwise disjoint?

„A family of sets is pairwise disjoint or mutually disjoint if every two different sets in the family are disjoint.“ – from Wikipedia article "Disjoint sets" What about the empty family of sets? Is ...
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0answers
19 views

Checking the definition of absolutely irreducible representations

The definition of an irreducible representation $(\rho, V) $ is one with no subrepresentations. Am I correct in saying that a absolutely irreducible means "it is irreducible over the algebraic ...
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1answer
47 views

Limit as universal arrow

I was interested in the definition of limit as universal arrow from $\Delta$ to $F$ given in Mac Lane's CWM book. Let us begin with some notation: Let $C$ be a category, and $J$ be an index ...
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1answer
39 views

What is the difference between a function and a functional?

When we read Functional Analysis it is said that it is a study of Functionals. I want to know how is it different from the study of functions.
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1answer
16 views

meaning of $f_{\chi_{E}}$

given $(X,\mathcal{M})$ a measurable space, I Have $E \subset X$ and $\chi_{E}$ is an indicator function. then what is meant by $f_{\chi_{E}}$ ? I am not very clear with this notation and meaning.
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55 views

Why is the conditional probability treated as a definition in Kolmogorov's probability theory?

The conditional probability is defined as: $$P(A|B) = \frac {P(A \cap B)} {P(B)}$$ given that $$P(B) \neq 0$$ This is achieved based on our intuition, along with the Venn diagram description of the ...
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18 views

“Branch” of a correspondence

I just saw for the first time references to the "branches" of a correspondence. Examples of the use of the terminology can be found in ...
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0answers
31 views

Does the relation $\mid^*$ have any interesting applications for understanding the structure of commutative rings that aren't integral domains?

There is a binary relation $\mid^*$ defined on any commutative ring as follows: $a \mid^* b$ iff $ak=b$ for some $k \in R$ that is not a zero divisor. This is always transitive, and it is reflexive ...
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1answer
43 views

Help with the definition of ordinal sum

"Given two ordinals $\alpha$ and $\beta$, let $A = (\alpha$ x {0}) $\cup$ $(\beta$ x {1}). Then, define a well ordering on A by: $(v, i) <_{A} (\tau, j) \iff (i \lt j) \lor (i = j$ $\land$ $v ...
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0answers
30 views

Definition - limit of a sequence - uses “rank”?

I have the following definition in my book, and was confused as to the context of the word "rank" here. The definition is as follows: A sequence $(u_n)_{n∈N}$ has limit $l ∈ R$ as $n → ∞$ (we also ...
3
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1answer
52 views

What is a endomorphism of vector bundle?

Quick question: When we say $f:E\to E$ is an endomorphism of the vector bundle $\pi:E\to M$, do we require that $f$ maps each fiber $E_p$ to itself, or it could be to another fiber $E_q$? I couldn't ...
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3answers
207 views

Is there an area of study regarding why certain mathematical definitions are useful?

Often in my studies I'll come across an definition, which I understand, and but don't necessarily see why the particular definition was chosen to be studied. For example, the topological axioms ...
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1answer
51 views

Is it a definition of cotangent?

$\cot(\alpha)$ - a value of $x$ coordinate of a point of intersection of $y=1$ and a line passing through $(0,0)$ and a point rotated $\alpha$ radians counterclockwise on the unit circle from ...
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1answer
13 views

Is it okay to define k-th symmetric power of $M$ in this way?

I want to define the tensor algebra and related algebras in a very formal way. I will illustrate how I tried to define algebras below. Let $R$ be a commutative ring and $M$ be an $R$-module. ...
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0answers
25 views

A quick question on the limsup of sets and logical connectives

I know we can interpret $x\in\limsup{A_n}$ iff $x$ is in infinitely many of the $A_n$'s, but am confused as to which logical connective we need to use to describe when $x\in\limsup{A_n}$, when we ...
3
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1answer
52 views

Subtlety in the Definition of Limit Point

In my time studying mathematics I have always found some subtle confusion with the definition of a limit point. I know it's possible for different definitions to yield the same results, and the ...
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0answers
19 views

What's actually $S^k(M)$?

Let $R$ be a commutative ring and $M$ be an $R$-module. Let $T(M)$ be the tensor algebra of $M$. Then, what is $S(M)$ (symmetric algebra and $S^k(M)$? Some articles define $S(M)$ as a quotient of ...
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0answers
28 views

Unique Union Problem

Given a the set $a = \{1,...,n\}$ and let $b$ denote a set of subsets of $a$. Find a subset of $c$ of $b$ so that the union of all subsets in $c$ is equal to $a$ and the intersection of any of the ...
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2answers
22 views

definition of negative binomial in probability karr

The book defines the probability of the negative binomial as: $$P\{X=k\}={{k-1}\choose{n-1}} p^k (1-p)^{k-n}$$ but where does the ${k-1}\choose{n-1}$ come from? It's quite different to wikipedia's ...
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0answers
24 views

Definition of some terms of transformation geometry

Recently I was studying transformation geometry from problem solving strategies . I liked the subject but could not understand some terms. Please anyone help me--- What is isometry? What is ...
2
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1answer
51 views

How to call “equivalent-looking” vertices in graph..?

In the above figure, the vertices expressed as blue dots are "equivalent-looking." Although my expression is somewhat ambiguous, I believe one can simply answer it. How can we call such vertices? ...
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1answer
23 views

Intuition behind a particular definition regarding cycles of partitions

I'm reading this paper on cycles of partitions, and was wondering if anyone could motivate the last condition in the definition of the sets $M_n$ in terms of the partitions being examined. In ...
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0answers
40 views

What is the “minimal” structure in which points and lines are defined?

Points, straight lines and planes are fundamental concept of geometry. Usually this entities are defined in a structure. We can easily define points in a vector space, or in a affine or projective ...
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2answers
56 views

how to understand the definition of continuity in analysis?

Please have a look at the picture above. This is about the continuity in analysis. I don't really understand how to utilize this definition? It says that is statement is equivalent to f is ...
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1answer
35 views

How to prove that homology is a functor?

Is a homology operator $H_k:(cKom) \rightarrow (Ab)$ a functor? I know this is a really simple question, but I'm not familiar with category theory and do not know how to prove this... (I'm even not ...
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2answers
176 views

Definition of a Minimal Set

A few times while studying math I have encountered the notion of a "minimal set". For example, given some set of subsets, what is the "minimal" sigma algebra generated by that set of subsets? Or, in ...
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1answer
22 views

interpreting the curve of intersection

I would like to understand the idea of a 'curve of intersection' in $\mathbb{R}^{3}$. Say we are given a surface $z = x^{\frac{1}{3}}y^{\frac{1}{3}}$ and a plane $y = x$. Then the curve of ...
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2answers
62 views

What's a local angle?

When I was trying to understand the definition of conformal map I got confused. A conformal map is a function $f: U \to \mathbb C$ where $U \subset \mathbb C$ such that $f$ preserves local angles. ...
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4answers
80 views

Find the derivative of $f(x) = \sec(x)$ without the quotient rule

Part of an analysis assignment I have: " Given $f(x) = \sec(x)$, compute the derivative of $f(x)$ by using the definition of derivative. (Note that $\sec(x) = 1/\cos(x)$ and $(\cos(x))' = ...
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2answers
40 views

About a norm : $p(uv)=p(u)p(v)$ all the time? [closed]

Say, $p$ is a norm on a vector space. Then can we say that $$p(uv)=p(u)p(v)$$ all the time? Thanks.